Logarithm of Infinite Product of Real Numbers
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Theorem
Let $(a_n)$ be a sequence of strictly positive real numbers.
Convergence
The following are equivalent:
- The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.
Divergence to zero
The following are equivalent:
- The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$.
Divergence to infinity
The following are equivalent:
- The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.